Question

Find \(v(t), a(t), \mathbf{T}(t),\) and \(N(t)\) (if it exists) for an object moving along the path given by the vector-valued function \(\mathrm{r}(t) .\) Use the results to determine the form of the path. Is the speed of the object constant or changing? $$ \mathbf{r}(t)=4 t \mathbf{i} $$

Short answer

The velocity function of the object is \(v(t) = 4\mathbf{i}\). The acceleration function of the object is \(a(t) = 0\). The unit tangent vector is \(\mathbf{T}(t) = \mathbf{i}\). The form of the path is a straight line along the x-axis. The speed of the object is constant.

Step-by-step solution

Calculate the Velocity Function

The velocity function \(v(t)\) is the derivative of the position function \(\mathbf{r}(t)\). Since \(\mathbf{r}(t) = 4t\mathbf{i}\), the derivative of \(\mathbf{r}(t)\) will give the velocity function \(v(t)\). That is, \(v(t) = r'(t) = 4\mathbf{i}\).

Calculate the Acceleration Function

The acceleration function \(a(t)\) is given by the derivative of the velocity function \(v(t)\). Since \(v(t) = 4\mathbf{i}\), its derivative is \(a(t) = v'(t) = 0\).

Calculate the Unit Tangent Vector

The unit tangent vector \(\mathbf{T}(t)\) at a point is given by the velocity vector at that point divided by the magnitude of the velocity. In this case, \(v(t) = 4\mathbf{i}\) and the magnitude of \(v(t)\) is 4. Therefore, \(\mathbf{T}(t) = v(t) / |v(t)| = 4\mathbf{i} / 4 = \mathbf{i}\).

Determine the Form of the Path

The path of the object can be determined by the position function \(\mathbf{r}(t) = 4t\mathbf{i}\). This indicates a straight line path along the x-axis.

Determine If Speed Is Constant Or Changing

Since the acceleration of the object is 0, the speed of the object is constant. The object is moving with a constant speed along the straight line path.

Key concepts

Velocity Function

In calculus, when we talk about an object's movement, the velocity function, often denoted as v(t), is a crucial concept to understand. This is essentially the rate at which the object's position is changing at any given time, t. To find the velocity, you differentiate the position function with respect to time, which in the context of vector-valued functions, is done component-wise.

Consider the exercise where \( \mathbf{r}(t) = 4t\mathbf{i} \). The velocity function is found by taking the derivative of \( \mathbf{r}(t) \) with respect to time, giving us \( v(t) = 4\mathbf{i} \). Notably, this function is constant, implying that the object moves at a steady pace. It does not speed up or slow down. The constant vector component \( \mathbf{i} \) signals motion along the x-axis.

Acceleration Function

Acceleration is the rate at which the velocity of an object changes over time, given by the function a(t). To obtain it, we differentiate the velocity function with respect to time. From our exercise, \( v(t) \) is constant, so its derivative, which we'd anticipate to give us the acceleration, is zero: \( a(t) = 0 \).

This finding indicates that the object is not accelerating; there are no forces acting on it to change its speed or direction. This concept is instrumental in predicting the future behavior of an object under certain forces or when it travels through different mediums which might affect its speed.

Unit Tangent Vector

The unit tangent vector, denoted typically as \( \mathbf{T}(t) \), is important as it provides directionality to the object's motion. It's a normalized vector that is tangent to the path of the object. When you normalize a vector, you scale the vector to have a magnitude of 1 but maintain its direction.

In our case, we have the velocity vector \( v(t) = 4\mathbf{i} \) with a magnitude of 4. To find the unit tangent vector, we divide the velocity by its magnitude, resulting in \( \mathbf{T}(t) = \mathbf{i} \). This tells us that, at any point along the object's path, it will be tangentially moving in the x-direction. The unit tangent vector is far more than just a direction; it's fundamental in defining the instantaneous directional behavior of the object.

Object Motion Analysis

Analyzing an object's motion involves looking at how its position, velocity, and acceleration changes over time. From the given exercise, we know that the object follows the path determined by the position function \( \mathbf{r}(t) = 4t\mathbf{i} \), and the object has a consistent velocity and zero acceleration. These factors indicate that the object is moving in a straight line along the x-axis.

Moreover, since the acceleration is zero, we deduced that the speed is constant. This kind of analysis is quintessential in physics and engineering, as it allows us to understand and predict the movement of objects in various conditions, which can be vital for planning and problem-solving in real-world applications.